Brief Announcement: Graph Exploration Using Constant-Size Memory and Storage

We consider the exploration problem in undirected graphs without node labels, which requires that a mobile agent initially placed at an arbitrary node visits all nodes and terminates. We assume that both of the agent and nodes are equipped with little memory, and the algorithm cannot use any initial knowledge on the topology of the graph. In this paper, we propose a new deterministic polynomial-time exploration (more precisely, depth-first search) algorithm which can be implemented using only O(1)-bit memory of the agent and O(1)-bit storage for each node. To the best of our knowledge, this is the first polynomial-time exploration algorithm achieving both sublogarithmic memory and sublogarithmic storage. The technical ingredient of our algorithm consists of the idea from the recent progress on small-space DFS algorithms by Asano et al. [ISAAC2014] and Elmasry et al. [STACS2015], and a new distributed backtrack algorithm for DFS paths. The algorithm also includes a new compact (i.e., using O(1)-bit storage) s-t path maintenance mechanism, which may be of independent interest.